---
tag: S2CV71FCY2
kind: definition
name: Suspension flow
msc: [37C10, 37A10, 37B02]
defines: MX16DQ5Q8G
uses: [0QT00PB5G2, 0X3WGX95FE, 0NQ1H2X495, AYVTR2ES5B]
sources:
  - url: https://en.wikipedia.org/wiki/Mapping_torus
---

## Definition

Let $X$ be a topological space, let $T \colon X \to X$ be a [[homeomorphism|0QT00PB5G2]], and let $r \colon X \to (0, \infty)$ be a [[continuous function|0X3WGX95FE]], the *roof function*, with $\inf_X r > 0$.
On $X \times \R$, let $\sim$ be the equivalence relation generated by
$$(x, s + r(x)) \sim (T(x), s) \quad \text{for all } x \in X,\ s \in \R.$$
Let $X_r = (X \times \R) / \sim$ with the [[quotient topology|0NQ1H2X495]], and write $[x, s]$ for the class of $(x, s)$.
The *suspension flow* of $T$ under $r$ is the flow $(\varphi_t)_{t \in \R}$ on $X_r$ given by
$$\varphi_t([x, s]) = [x, s + t].$$

## Remarks

The flow is well defined because the generating relation commutes with the translations $(x, s) \mapsto (x, s + t)$.
Every class has a representative $(x, s)$ with $0 \le s < r(x)$.
So $X_r$ can be pictured as the region under the graph of $r$, with the top point $(x, r(x))$ glued to the bottom point $(T(x), 0)$.
For $0 \le s < r(x)$ and $t \ge 0$, the point $\varphi_t([x, s])$ equals $[T^n(x), s + t - \sum_{k=0}^{n-1} r(T^k(x))]$, where $T^k$ is the $k$-th [[iterate|AYVTR2ES5B]] and $n \ge 0$ is the number of times the orbit has hit the roof.
The first-return map of the flow to the base $\set{[x, 0] : x \in X}$ is $T$, and the return time of $[x, 0]$ is $r(x)$.
If $\mu$ is a $T$-invariant measure, the product of $\mu$ with Lebesgue measure on the region under the graph of $r$ is invariant under the flow.

Example: let $X = \R / \Z$, $T(x) = x + \alpha \bmod 1$ and $r = 1$.
The map $[x, s] \mapsto (x + \alpha s, s) \bmod \Z^2$ is well defined, since $(x, s + 1)$ and $(x + \alpha, s)$ both go to $(x + \alpha s + \alpha, s) \bmod \Z^2$.
It identifies $X_r$ with the torus $\R^2 / \Z^2$, and it carries the suspension flow to the linear flow $(u, v) \mapsto (u + \alpha t, v + t)$ in the direction $(\alpha, 1)$.

The condition $\inf_X r > 0$ makes the sums $\sum_{k=0}^{n-1} r(T^k(x))$ and $\sum_{k=1}^{n} r(T^{-k}(x))$ tend to infinity, which is what gives every class a representative under the roof.
Non-example: the doubling map $x \mapsto 2x \bmod 1$ is not a homeomorphism, so the construction above does not apply to it; for non-invertible maps one uses a suspension semiflow, defined only for $t \ge 0$.
With $r = 1$, the space $X_r$ is the mapping torus of $T$.